math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.0% → 87.8%

Time: 5.6s

Alternatives: 5

Speedup: 2.0×

Specification

\[im > 0\]

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 116000000 \lor \neg \left(re \leq 7 \cdot 10^{+30}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.2e-62)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (if (or (<= re 116000000.0) (not (<= re 7e+30)))
     (* 0.5 (/ im (sqrt re)))
     (* 0.5 (sqrt (* 2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.2e-62) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else if ((re <= 116000000.0) || !(re <= 7e+30)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.2e-62) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else if ((re <= 116000000.0) || !(re <= 7e+30)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.2e-62:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	elif (re <= 116000000.0) or not (re <= 7e+30):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.2e-62)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	elseif ((re <= 116000000.0) || !(re <= 7e+30))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.2e-62)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	elseif ((re <= 116000000.0) || ~((re <= 7e+30)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.2e-62], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 116000000.0], N[Not[LessEqual[re, 7e+30]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.20000000000000017e-62

   1. Initial program 51.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 51.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 51.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 51.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 51.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-def 94.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 94.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   if 2.20000000000000017e-62 < re < 1.16e8 or 7.00000000000000042e30 < re

   1. Initial program 11.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. pow1/2 11.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{0.5}} \]

      2. hypot-udef 30.5%

         \[\leadsto 0.5 \cdot {\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)\right)}^{0.5} \]

      3. pow-to-exp 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   3. Applied egg-rr 29.0%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in im around 0 85.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 85.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]

      2. unpow-1 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-1}}} \cdot im\right) \]

      3. metadata-eval 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot im\right) \]

      4. pow-sqr 85.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \cdot im\right) \]

      5. rem-sqrt-square 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left|{re}^{-0.5}\right|} \cdot im\right) \]

      6. rem-square-sqrt 84.6%

         \[\leadsto 0.5 \cdot \left(\left|\color{blue}{\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}}\right| \cdot im\right) \]

      7. fabs-sqr 84.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}\right)} \cdot im\right) \]

      8. rem-square-sqrt 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{{re}^{-0.5}} \cdot im\right) \]

      9. exp-to-pow 79.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{e^{\log re \cdot -0.5}} \cdot im\right) \]

      10. metadata-eval 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\log re \cdot \color{blue}{\left(-0.5\right)}} \cdot im\right) \]

      11. distribute-rgt-neg-in 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-\log re \cdot 0.5}} \cdot im\right) \]

      12. exp-neg 79.6%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{e^{\log re \cdot 0.5}}} \cdot im\right) \]

      13. exp-to-pow 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{{re}^{0.5}}} \cdot im\right) \]

      14. unpow1/2 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{\sqrt{re}}} \cdot im\right) \]

      15. associate-*l/ 85.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot im}{\sqrt{re}}} \]

      16. *-lft-identity 85.1%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}} \]

   6. Simplified 85.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 1.16e8 < re < 7.00000000000000042e30

   1. Initial program 42.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   3. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   4. Simplified 100.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 116000000 \lor \neg \left(re \leq 7 \cdot 10^{+30}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 2: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000385:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-62} \lor \neg \left(re \leq 980000000\right) \land re \leq 3 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000385)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 2.1e-62) (and (not (<= re 980000000.0)) (<= re 3e+25)))
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000385) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 2.1e-62) || (!(re <= 980000000.0) && (re <= 3e+25))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000385d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 2.1d-62) .or. (.not. (re <= 980000000.0d0)) .and. (re <= 3d+25)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000385) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 2.1e-62) || (!(re <= 980000000.0) && (re <= 3e+25))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000385:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 2.1e-62) or (not (re <= 980000000.0) and (re <= 3e+25)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000385)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 2.1e-62) || (!(re <= 980000000.0) && (re <= 3e+25)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000385)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 2.1e-62) || (~((re <= 980000000.0)) && (re <= 3e+25)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000385], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.1e-62], And[N[Not[LessEqual[re, 980000000.0]], $MachinePrecision], LessEqual[re, 3e+25]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.8499999999999998e-4

   1. Initial program 46.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around -inf 80.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 80.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.8499999999999998e-4 < re < 2.0999999999999999e-62 or 9.8e8 < re < 3.00000000000000006e25

   1. Initial program 54.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 77.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   3. Step-by-step derivation

      1. *-commutative 77.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   4. Simplified 77.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 2.0999999999999999e-62 < re < 9.8e8 or 3.00000000000000006e25 < re

   1. Initial program 11.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. pow1/2 11.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{0.5}} \]

      2. hypot-udef 30.5%

         \[\leadsto 0.5 \cdot {\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)\right)}^{0.5} \]

      3. pow-to-exp 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   3. Applied egg-rr 29.0%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in im around 0 85.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 85.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]

      2. unpow-1 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-1}}} \cdot im\right) \]

      3. metadata-eval 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot im\right) \]

      4. pow-sqr 85.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \cdot im\right) \]

      5. rem-sqrt-square 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left|{re}^{-0.5}\right|} \cdot im\right) \]

      6. rem-square-sqrt 84.6%

         \[\leadsto 0.5 \cdot \left(\left|\color{blue}{\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}}\right| \cdot im\right) \]

      7. fabs-sqr 84.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}\right)} \cdot im\right) \]

      8. rem-square-sqrt 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{{re}^{-0.5}} \cdot im\right) \]

      9. exp-to-pow 79.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{e^{\log re \cdot -0.5}} \cdot im\right) \]

      10. metadata-eval 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\log re \cdot \color{blue}{\left(-0.5\right)}} \cdot im\right) \]

      11. distribute-rgt-neg-in 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-\log re \cdot 0.5}} \cdot im\right) \]

      12. exp-neg 79.6%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{e^{\log re \cdot 0.5}}} \cdot im\right) \]

      13. exp-to-pow 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{{re}^{0.5}}} \cdot im\right) \]

      14. unpow1/2 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{\sqrt{re}}} \cdot im\right) \]

      15. associate-*l/ 85.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot im}{\sqrt{re}}} \]

      16. *-lft-identity 85.1%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}} \]

   6. Simplified 85.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000385:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-62} \lor \neg \left(re \leq 980000000\right) \land re \leq 3 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 600000000 \lor \neg \left(re \leq 6.5 \cdot 10^{+26}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.1e+36)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.6e-63)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 600000000.0) (not (<= re 6.5e+26)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.1e+36) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.6e-63) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 600000000.0) || !(re <= 6.5e+26)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.1d+36)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.6d-63) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 600000000.0d0) .or. (.not. (re <= 6.5d+26))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.1e+36) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.6e-63) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 600000000.0) || !(re <= 6.5e+26)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.1e+36:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.6e-63:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 600000000.0) or not (re <= 6.5e+26):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.1e+36)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.6e-63)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 600000000.0) || !(re <= 6.5e+26))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.1e+36)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.6e-63)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 600000000.0) || ~((re <= 6.5e+26)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.1e+36], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-63], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 600000000.0], N[Not[LessEqual[re, 6.5e+26]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.10000000000000004e36

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around -inf 85.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 85.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 85.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -2.10000000000000004e36 < re < 5.6000000000000005e-63

   1. Initial program 56.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 75.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]

   if 5.6000000000000005e-63 < re < 6e8 or 6.50000000000000022e26 < re

   1. Initial program 11.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. pow1/2 11.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{0.5}} \]

      2. hypot-udef 30.5%

         \[\leadsto 0.5 \cdot {\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)\right)}^{0.5} \]

      3. pow-to-exp 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   3. Applied egg-rr 29.0%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in im around 0 85.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 85.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]

      2. unpow-1 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-1}}} \cdot im\right) \]

      3. metadata-eval 85.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot im\right) \]

      4. pow-sqr 85.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \cdot im\right) \]

      5. rem-sqrt-square 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left|{re}^{-0.5}\right|} \cdot im\right) \]

      6. rem-square-sqrt 84.6%

         \[\leadsto 0.5 \cdot \left(\left|\color{blue}{\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}}\right| \cdot im\right) \]

      7. fabs-sqr 84.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}\right)} \cdot im\right) \]

      8. rem-square-sqrt 85.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{{re}^{-0.5}} \cdot im\right) \]

      9. exp-to-pow 79.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{e^{\log re \cdot -0.5}} \cdot im\right) \]

      10. metadata-eval 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\log re \cdot \color{blue}{\left(-0.5\right)}} \cdot im\right) \]

      11. distribute-rgt-neg-in 79.6%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-\log re \cdot 0.5}} \cdot im\right) \]

      12. exp-neg 79.6%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{e^{\log re \cdot 0.5}}} \cdot im\right) \]

      13. exp-to-pow 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{{re}^{0.5}}} \cdot im\right) \]

      14. unpow1/2 84.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{\sqrt{re}}} \cdot im\right) \]

      15. associate-*l/ 85.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot im}{\sqrt{re}}} \]

      16. *-lft-identity 85.1%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}} \]

   6. Simplified 85.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 6e8 < re < 6.50000000000000022e26

   1. Initial program 42.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   3. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   4. Simplified 100.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 600000000 \lor \neg \left(re \leq 6.5 \cdot 10^{+26}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 4: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e-6) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-6) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.2d-6)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-6) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e-6:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e-6)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e-6)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e-6], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -7.19999999999999967e-6

   1. Initial program 46.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around -inf 80.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 80.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -7.19999999999999967e-6 < re

   1. Initial program 37.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 55.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   3. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   4. Simplified 55.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 5: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* 2.0 im))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * im));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * im))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.3%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

2. Taylor expanded in re around 0 45.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
3. Step-by-step derivation

   1. *-commutative 45.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

4. Simplified 45.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

5. Final simplification 45.7%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]

Reproduce

herbie shell --seed 2023339 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))